Math 151
Sections 5.2 and 5.3
Maximum and Minimum Values
Derivatives and the Shapes of Curves
Recall that a critical number (critical value) is a number, c, in the domain of f such that f ′(c) = 0 or
f ′(c) does not exist.
Extreme Values and the First Derivative Test
If f has a local extremum (local maximum or minimum) at c then c is a critical value of f (x).
Alternatively, let c be a critical number of a continuous function f.
• If f ! changes from positive to negative at c, then f has a local maximum at c.
• If f ! changes from negative to positive at c, then f has a local minimum at c.
• If f ! does not change sign at c, then f has no local maximum or minimum at c.
Example: Find the intervals where the function is increasing and the intervals where it is
decreasing. Classify all critical values.
A. y = x 3 + 3x 2 !9x + 8
B. y = 3x 5 ! 20x 3 + 20
C. y =
x 2 +1
x
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D. y = ( x 2 !16)3
E. y = x ln x
F. y = xe x
2 !3x
G. y ! = ( x " 4) ( x + 2) , and the domain of y is all real numbers
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Math 151
Absolute Maxima and Minima
A function f has an absolute maximum at c if f (c) ! f ( x) for all x in D, where D is the domain of
f. f (c) is called the maximum value of f on D.
Similarly, f has an absolute minimum at c if f (c) ! f ( x) for all x in D and the number f (c) is
called the minimum value of f on D.
The maximum and minimum values of f are called the extreme values of f.
Example: For these functions, find the absolute max and the absolute min, if they exist.
A. y = 3x 2 ! x 3 +1
B. y = x 4 ! 4x 3
Restricted Domains
Math 151
Extreme Values on a Closed Interval If f is a continuous function on a closed interval [a, b], then
f will have both an absolute max and an absolute min. The extreme values will be located at critical
values in the interval or at the endpoints of the interval, x = a or x = b.
Example: For the function f ( x) = 12x 2 !3x 3 +1 , find the absolute max and the absolute min on
the indicated interval.
A. [2, 5]
B. [−3, 5]
C. (−3, 5]
Example: For the function f ( x) =
1
( x ! 4)
2
, find the absolute max and the absolute min on the
interval [0, 5].
Example: For the function f ( x) = cos x , find the absolute max and the absolute min on the interval
" ! ! %'
$! , '' .
$ 2 2 '&
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Math 151
Inflection Points
x = c in the domain of f is a possible inflection point (PIP) if f ′′(c) = 0 or f ′′(c) does not exist.
If f changes concavity at c then c is a point of inflection of f (x).
Example: Find the intervals where the function is concave up and the intervals where it is concave
down. Find the x-coordinate of the inflection points.
A. y = x 5 !5x 4 +10x + 5
B. y = x ln ( x ! 2)
Example: Find the values of a and b such that f ( x) = ax 2 ! bln x will have an inflection point at (1, 5).
Math 151
The Second Derivative Test Let c be a PIP of a continuous function f and let f ′′ be continuous near c.
• If f !(c) = 0 and f !!(c) > 0 , then f has a local minimum at c.
•
If f !(c) = 0 and f !!(c) < 0 , then f has a local maximum at c.
Example: Suppose that f has critical values of x = 0, x = 2, and x = −2. If f !!( x) = 60x 3 "120x ,
what conclusion can be drawn about the critical values?
Example: For y = e!x , identify the following and sketch the graph.
• Roots, points of discontinuity, VA, HA
• Intervals of increasing or decreasing
• Relative and absolute maxima and minima
• Intervals of concave up or concave down
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Math 151
The Mean Value Theorem If f is a differentiable function on the interval [a, b], then there exists a
number c between a and b such that
f (b) " f ( a)
f ! ( c) =
b" a
Example: Find a number c that satisfies the conclusion of the Mean Value Theorem on the interval
[0, 2] for f ( x) = x 3 + x !1 .
Example: You enter a toll road at 8am and then exit it at 9:15am. The distance between the entrance
and exit is 100 miles. If the speed limit is set at 70mph, could you be fined for speeding?